A G_2 Unification of the Deformed and Resolved Conifolds

TL;DR

This paper derives unified first-order equations for G_2 metrics with S^3×S^3 symmetry, connecting known conifold-related solutions and introducing new ALC G_2 metrics with rich geometric structures.

Contribution

It presents a unified system of first-order equations for G_2 metrics, linking deformed and resolved conifold geometries and discovering new ALC solutions with novel topologies.

Findings

Unified G_2 metrics encompass deformed and resolved conifolds.

New ALC G_2 solutions with  topology and T^{1,1} bolt.

Weak-coupling limits relate to Calabi-Yau metrics on line bundles.

Abstract

We find general first-order equations for G_2 metrics of cohomogeneity one with S^3\times S^3 principal orbits. These reduce in two special cases to previously-known systems of first-order equations that describe regular asymptotically locally conical (ALC) metrics \bB_7 and \bD_7, which have weak-coupling limits that are S^1 times the deformed conifold and the resolved conifold respectively. Our more general first-order equations provide a supersymmetric unification of the two Calabi-Yau manifolds, since the metrics \bB_7 and \bD_7 arise as solutions of the {\it same} system of first-order equations, with different values of certain integration constants. Additionally, we find a new class of ALC G_2 solutions to these first-order equations, which we denote by \wtd\bC_7, whose topology is an \R^2 bundle over T^{1,1}. There are two non-trivial parameters characterising the homogeneous squashing of the T^{1,1} bolt. Like the previous examples of the \bB_7 and \bD_7 ALC metrics, here too there is a U(1) isometry for which the circle has everywhere finite and non-zero length. The weak-coupling limit of the \wtd\bC_7 metrics gives S^1 times a family of Calabi-Yau metrics on a complex line bundle over S^2\times S^2, with an adjustable parameter characterising the relative sizes of the two S^2 factors.